lecture 5: linear and affine transformations
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Linear transformationsAffine transformationsTransformations in 3D
Graphics 2011/2012, 4th quarter
Lecture 5
Linear and affine transformations
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
DefinitionExamplesFinding matricesCompositions of transformationsTransposing normal vectors
Vector transformation: basic idea
Multiplication of an n× n matrix with a vector(i.e. a n× 1 matrix):
In 2D:
(a11 a12
a21 a22
)(xy
)=(a11x+ a12ya21x+ a22y
)
In 3D:
a11 a12 a13
a21 a22 a23
a31 a32 a33
xyz
=
a11x+ a12y + a13za21x+ a22y + a23za31x+ a32y + a33z
The result is a (transformed) vector.
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
DefinitionExamplesFinding matricesCompositions of transformationsTransposing normal vectors
Example: scaling
To scale with a factor two with respectto the origin, we apply the matrix(
2 00 2
)to a vector:(
2 00 2
)(xy
)=(
2x+ 0y0x+ 2y
)=(
2x2y
)
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
DefinitionExamplesFinding matricesCompositions of transformationsTransposing normal vectors
Linear transformations
A function T : Rn → Rm is called alinear transformation if it satisfies
1 T (~u+ ~v) = T (~u) + T (~v) for all~u,~v ∈ Rn.
2 T (c~v) = cT (~v) for all ~v ∈ Rn andall scalars c.
Or (alternatively), if it satisfies
T (c1~u+ c2~v) = c1T (~u) + c2T (~v)for all ~u,~v ∈ Rn and all scalars c1, c2.
u
v
u + v
T (u)
T (v)
T (u + v)
Linear transformations can be represented by matrices
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
DefinitionExamplesFinding matricesCompositions of transformationsTransposing normal vectors
Example: scaling
We already saw scaling by a factor of 2.
In general, a matrix(a11 00 a22
)scales the i-th coordinate of a vector by the factor aii 6= 0, i.e.(a11 00 a22
)(xy
)=(a11x+ 0y0x+ a22y
)=(a11xa22y
)
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
DefinitionExamplesFinding matricesCompositions of transformationsTransposing normal vectors
Example: scaling
Scaling does not have to beuniform. Here, we scale with afactor one half in x-direction,and three in y-direction:(
12 00 3
)(xy
)=(
12x3y
)Q: what is the inverse of thismatrix?
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
DefinitionExamplesFinding matricesCompositions of transformationsTransposing normal vectors
Example: scaling
Using Gaussian elimination tocalculate the inverse of amatrix:
(12 0 1 00 3 0 1
)
(Gaussian elimination)
(1 0 2 00 1 0 1
3
)Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
DefinitionExamplesFinding matricesCompositions of transformationsTransposing normal vectors
Example: projection
We can also use matrices todo orthographic projections,for instance, onto the Y -axis:(
0 00 1
)(xy
)=(
0y
)Q: what is the inverse of thismatrix?
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
DefinitionExamplesFinding matricesCompositions of transformationsTransposing normal vectors
Example: reflection
Reflection in the line y = xboils down to swapping x- andy-coordinates:(
0 11 0
)(xy
)=(yx
)Q: what is the inverse of thismatrix?
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
DefinitionExamplesFinding matricesCompositions of transformationsTransposing normal vectors
Example: shearing
Shearing in x-direction“pushes things sideways”:(
1 10 1
)(xy
)=(x+ yy
)We can also “push thingsupwards” with(
1 01 1
)(xy
)=(
xx+ y
)
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
DefinitionExamplesFinding matricesCompositions of transformationsTransposing normal vectors
Example: shearing
General case for shearing inx-direction:(
1 s0 1
)(xy
)=(x+ syy
)And its inverse operation:(
1 −s0 1
)(xy
)=(x− syy
)
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
DefinitionExamplesFinding matricesCompositions of transformationsTransposing normal vectors
Example: rotation
To rotate 45◦ about the origin, weapply the matrix(√
22 −
√2
2√2
2
√2
2
)=√
22
(1 −11 1
)
Note:√
22 = cos 45◦ = sin 45◦,
so this is the same as(cos 45◦ − sin 45◦
sin 45◦ cos 45◦
)
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
DefinitionExamplesFinding matricesCompositions of transformationsTransposing normal vectors
Example: rotation
General case for a counterclockwiserotation about an angle φ around theorigin:(
cosφ − sinφsinφ cosφ
)And for clockwise rotation:(
cosφ sinφ− sinφ cosφ
)
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
DefinitionExamplesFinding matricesCompositions of transformationsTransposing normal vectors
Finding matrices
Applying matrices is prettystraightforward, but how do wefind the matrix for a givenlinear transformation?
Let A =(a11 a12
a21 a22
)Q: what values do we have toassign to the aij ’s to achieve acertain transformation?
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
DefinitionExamplesFinding matricesCompositions of transformationsTransposing normal vectors
Finding matrices
Let’s apply some transformations to the base vectors ~b1 = (1, 0) and~b2 = (0, 1) of the carthesian coordinates system, e.g.
Scaling (factors a, b 6= 0):
(a 00 b
)(xy
)=
(axby
)
Shearing (x-dir., factor s):
(1 s0 1
)(xy
)=
(x + sy
y
)
Reflection (in line y = x):
(0 11 0
)(xy
)=
(yx
)
Rotation (countercl., 45◦):√
22
(1 −11 1
)(xy
)=√
22
(x− yx + y
)
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
DefinitionExamplesFinding matricesCompositions of transformationsTransposing normal vectors
Finding matrices
Aha! The column vectors of atransformation matrix are theimages of the base vectors!
That gives us an easy methodof finding the matrix for agiven linear transformation.
Why? Because matrixmultiplication is a lineartransformation.
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
DefinitionExamplesFinding matricesCompositions of transformationsTransposing normal vectors
Finding matrices
Remember: T is a linear transformation if and only if
T (c1~u+ c2~v) = c1T (~u) + c2T (~v)
Let’s look at carthesian coordinates, where each vector ~w can berepresented as a linear combination of the base vectors ~b1, ~b2:
~w =(xy
)= x
(10
)+ y
(01
)If we apply a linear transformation T to this vector, we get:
T ((xy
)) = T (x
(10
)+ y
(01
)) = xT (
(10
)) + yT (
(01
))
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
DefinitionExamplesFinding matricesCompositions of transformationsTransposing normal vectors
Finding matrices: example
Rotation (counterclockwise, angle φ):
(cosφ − sinφsinφ cosφ
)For example for the first base vector ~b1:
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
DefinitionExamplesFinding matricesCompositions of transformationsTransposing normal vectors
Example: reflection and scaling
Multiple transformations can be combinedinto one.Here, we first do a reflection in the liney = x, and then we scale with a factor 5 inx-direction, and a factor 2 in y-direction:(
5 00 2
)(0 11 0
)(xy
)=(
0 52 0
)(xy
)
Remember: matrix multiplication isassociative, i.e. A(Bx) = (AB)x.
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
DefinitionExamplesFinding matricesCompositions of transformationsTransposing normal vectors
Example: reflection and scaling
But: Matrix multiplication is not commutative,i.e. in general AB 6= BA, for example:(
5 00 2
)(0 11 0
)(xy
)=(
0 52 0
)(xy
)=(
5y2x
)(
0 11 0
)(5 00 2
)(xy
)=(
0 25 0
)(xy
)=(
2y5x
)Mind the order!
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
DefinitionExamplesFinding matricesCompositions of transformationsTransposing normal vectors
Transposing normal vectors
Unfortunately, normal vectors are notalways transformed properly.
E.g. look at shearing, where tangentvectors are correctly transformed butnormal vectors not.
To transform a normal vector ~ncorrectly under a given lineartransformation A, we have to applythe matrix (A−1)T . Why?
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
DefinitionExamplesFinding matricesCompositions of transformationsTransposing normal vectors
Transposing normal vectors
We know that tangent vectors are tranformed correctly: A~t = ~tA.
But this is not necessarily true for normal vectors: A~n 6= ~nA.
Goal: find matrix NA that transforms ~n correctly, i.e. NA~n = ~nN
where ~nN is the correct normal vector of the transformed surface.
We know that~nT~t = 0.
Hence, we can also say that~nT I~t = 0
which is is the same as~nTA−1A~t = 0
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
DefinitionExamplesFinding matricesCompositions of transformationsTransposing normal vectors
Transposing normal vectors
Because A~t = ~tA is our correctly transformed tangent vector,we have
~nTA−1 ~tA = 0
Because their scaler product is 0, ~nTA−1 must be orthogonal to it.So, the vector we are looking for must be
~nTN = ~nTA−1.
Because of how matrix multiplication is defined,this is a transposed vector. But we can rewrite this to
~nN = (~nTA−1)T .
And if you remember that (AB)T = BTAT , we get
~nN = (A−1)T~n
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Problem. . .. . . and solution: homogeneous coordinates
More complex transformations
So now we know how todetermine matrices for a giventransformation. Let’s tryanother one:
Q: what is the matrix for arotation of 90◦ about thepoint (2, 1)?
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Problem. . .. . . and solution: homogeneous coordinates
More complex transformations
We can form ourtransformation by composingthree simpler transformations:
Translate everything suchthat the center of rotationmaps to the origin.
Rotate about the origin.
Revert the translationfrom the first step.
Q: but what is the matrix for atranslation?
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Problem. . .. . . and solution: homogeneous coordinates
More complex transformations
Translation is not a linear transformation.
With linear transformations we get:
Ax =(a11 a12
a21 a22
)(xy
)=(a11x + a12ya21x + a22y
)But we need something like:(
x + xt
y + yt
)We can do this with a combination of linear transformations andtranslations called affine transformations.
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Problem. . .. . . and solution: homogeneous coordinates
Homogeneous coordinates
Observation:Shearing in 2D smells a lot liketranslation in 1D . . .
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Problem. . .. . . and solution: homogeneous coordinates
Homogeneous coordinates
. . . and shearing in3D smells liketranslation in 2D
(. . . and shearing in4D . . . )
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Problem. . .. . . and solution: homogeneous coordinates
Homogeneous coordinates in 2D: basic idea
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Problem. . .. . . and solution: homogeneous coordinates
Homogeneous coordinates in 2D: basic idea
We see: by adding a 3rd dimension to our 2D space, we can add aconstant value to the first two coordinates by matrixmultiplication.
M
xyd
=
x + xt
y + yt
d
That’s exactly what we want (for the first 2 coordinates).
But how does the matrix M look like?And how are we dealing with this 3rd coordinate?
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Problem. . .. . . and solution: homogeneous coordinates
Homogeneous coordinates
Shearing in 3D based on the z-coordinate is a simple generalizationof 2D shearing:1 0 xt
0 1 yt
0 0 1
xyd
=
x + xtdy + ytd
d
Notice that we didn’t make any assumption about d, . . .
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Problem. . .. . . and solution: homogeneous coordinates
Homogeneous coordinates: matrices
. . . so we can use, for example, d = 1:1 0 xt
0 1 yt
0 0 1
xy1
=
x + xt
y + yt
1
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Problem. . .. . . and solution: homogeneous coordinates
Homogeneous coordinates: points
Translations in 2D can be represented as shearing in 3D by lookingat the plane z = 1.
By representing all our 2D points (x, y) by 3D vectors (x, y, 1), wecan translate them about (xt, yt) using the following 3D shearingmatrix: 1 0 xt
0 1 yt
0 0 1
xy1
=
x+ xt
y + yt
1
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Problem. . .. . . and solution: homogeneous coordinates
Homogeneous coordinates: vectors
But can we translate a vector? No!(Remember: vectors are defined by their length and direction.)
Hence, we have to represent a vector differently, i.e. by (x, y, 0):1 0 xt
0 1 yt
0 0 1
xy0
=
xy0
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Problem. . .. . . and solution: homogeneous coordinates
Homogeneous coordinates
Affine transformations (i.e. linear transformations and translations)can be done with simple matrix multiplication if we addhomogeneous coordinates, i.e. (in 2D):
a third coordinate z = 1
to each location:
xy1
a third coordinate z = 0
to each vector:
xy0
a third row (0 . . . 0 1) toeach matrix:
∗.
. . . ..∗
0 . . . 0 1
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Problem. . .. . . and solution: homogeneous coordinates
Affine transformations: points vs. vectors
Scaling and moving a location/point (or “an object”):1 0 xt
0 1 yt
0 0 1
s1 0 00 s2 00 0 1
xy1
=
s1 0 xt
0 s2 yt
0 0 1
xy1
=
s1x + xt
s2y + yt
1
With the same matrix, we scale (but not move!) a vector:s1 0 xt
0 s2 yt
0 0 1
xy0
=
s1xs2y0
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Problem. . .. . . and solution: homogeneous coordinates
Affine transformations: example
What is the matrix forreflection in the liney = −x+ 5?
Idea: move everything to theorigin, reflect, and the moveeverything back.
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Problem. . .. . . and solution: homogeneous coordinates
Affine transformations: example
1. Translation of y = −x + 5 to y = −x1 0 xt
0 1 yt
0 0 1
xy1
=
x + xt
y + yt
1
2. Reflection on y = −x:(
0 −1−1 0
)
3. Translate y = −x back to y = −x + 5
1 0 .0 1 .0 0 1
0 −1 .−1 0 .
. . .
. . .. . .. . .
1 0 .
0 1 .0 0 1
. . .. . .. . .
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Problem. . .. . . and solution: homogeneous coordinates
Affine transformations
So, the matrix for reflection inthe line y = −x+ 5 is 0 −1 5−1 0 50 0 1
But what is the significance ofthe columns of the matrix?
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Problem. . .. . . and solution: homogeneous coordinates
Affine transformations
The last column vector is the imageof the origin after applying the affinetransformation
(Remember: The other columns arethe image of the base vectors underthe linear transformation)
(Also remember: The coordinates inthe last row are called homogeneouscoordinates)
0 −1 5−1 0 50 0 1
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Moving up a dimensionRotations in 3DReflections in 3D
Linear transformations in 3D
In general, linear transformations in 3D are a straightforwardextension of their 2D counterpart.
We have already seen an example: shearing in x− y−direction1 0 dx
0 1 dy
0 0 1
Or in general 1 dx2 dx3
dy1 1 dy3
0 0 1
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Moving up a dimensionRotations in 3DReflections in 3D
Affine transformations in 3D
Homogeneous coordinates (and therewith affine transformations)in 3D are also a straightforward generalization from 2D.
We just have to use 4× 4 matrices now:
Matrices:
a11 a12 a13 0a21 a22 a23 0a31 a32 a33 00 0 0 1
points:
xyz1
and vectors:
xyz0
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Moving up a dimensionRotations in 3DReflections in 3D
Transformations in 3D
For scaling, we have three scaling factors on the diagonal of thematrix. sx 0 0
0 sy 00 0 sz
Shearing can be done in either x-, y-, or z-direction (or acombination thereof). For example, shearing in x-direction:1 dx2 dx3
0 1 00 0 1
xyz
=
x+ dyy + dzzyz
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Moving up a dimensionRotations in 3DReflections in 3D
Transformations in 3D
We see: transformations in 3D are very similar to those in 2D.
Scaling and shearing are straightforward generalizations of the2D cases.
Reflection is done with respect to planes
Rotation is done about directed lines.
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Moving up a dimensionRotations in 3DReflections in 3D
Transformations in 3D: rotations
Q: What is the matrix for a rotation of angle φ about the z-axis?
Rotation in 2D (x− y-plane):(cosφ − sinφsinφ cosφ
)
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Moving up a dimensionRotations in 3DReflections in 3D
Transformations in 3D: rotations
Q: What is the matrix for a rotation of angle φ about the z-axis?
Rotation in 2D (x− y-plane):(cosφ − sinφsinφ cosφ
) Rotation in 3D around z-axis:cosφ − sinφ 0sinφ cosφ 0
0 0 1
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Moving up a dimensionRotations in 3DReflections in 3D
Transformations in 3D: rotations
Q: What is the matrix for a rotation of angle φ about the directedline (0, 1, 2) + t(3, 0, 4) t > 0?
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Moving up a dimensionRotations in 3DReflections in 3D
Transformations in 3D: rotations
We need a 3D transformationthat rotates around anarbitrary vector ~w.How can we do that?
Idea:
1 Rotate vector ~w to z-axis
2 Do 3D rotation aroundz-axis
3 Rotate everything back tooriginal position
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Moving up a dimensionRotations in 3DReflections in 3D
Transformations in 3D: rotations
We already know how to rotate around the z-axis:cosφ − sinφ 0sinφ cosφ 0
0 0 1
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Moving up a dimensionRotations in 3DReflections in 3D
Transformations in 3D: rotations
But how do we get the other 2 rotation matrices? ?
cosφ − sinφ 0sinφ cosφ 0
0 0 1
?
Rotating everything back to it’s original position after we havedone the rotation is easy.
Remember that the columns of the rotation matrix represent theimages of the carthesian basis vectors under the transformation!xu xv xw
yu yv yw
zu zv zw
cosφ − sinφ 0sinφ cosφ 0
0 0 1
?
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Moving up a dimensionRotations in 3DReflections in 3D
Transformations in 3D: rotations
Now all we still need is the original rotation matrixxu xv xw
yu yv yw
zu zv zw
cosφ − sinφ 0sinφ cosφ 0
0 0 1
?
But that’s just the inverse of the other rotation matrix.
Note: if we assume that ‖~u‖ = ‖~v‖ = ‖~w‖ = 1 we don’t even needto calculate it, because the inverse of an orthogonal matrix isalways its transposed!xu xv xw
yu yv yw
zu zv zw
cosφ − sinφ 0sinφ cosφ 0
0 0 1
xu yu zuxv yv zvxw yw zw
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Moving up a dimensionRotations in 3DReflections in 3D
Transformations in 3D: rotations
Hmm, but we only have the vector ~w. How do we get a wholecoordinates system ~u, ~v, ~w?
w
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Moving up a dimensionRotations in 3DReflections in 3D
Transformations in 3D: rotations
Notice that such a rotation is not unique. (And it doesn’t have tobe as long as the two rotation matrices are correct.)
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Moving up a dimensionRotations in 3DReflections in 3D
Transformations in 3D: reflections
Q: What is the matrix forreflection in XY -plane?
What happens to a randompoint (x, y, z) in 3D when wereflect it on the XY -plane?
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Moving up a dimensionRotations in 3DReflections in 3D
Transformations in 3D: reflections
Hence, the matrix forreflection in XY -plane is:
1 0 00 1 00 0 −1
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Moving up a dimensionRotations in 3DReflections in 3D
Transformations in 3D: reflections
Q: What is the matrix forreflection in XY -plane?
Q: What is the matrix forreflection in the plane3x+ 4y − 12z = 0?
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Moving up a dimensionRotations in 3DReflections in 3D
Transformations in 3D: reflections
Q: What is the matrix forreflection in XY -plane?
Q: What is the matrix forreflection in the plane3x+ 4y − 12z = 0?
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
Linear transformationsAffine transformationsTransformations in 3D
Moving up a dimensionRotations in 3DReflections in 3D
Transformations in 3D: reflections
Q: What is the matrix forreflection in XY -plane?
Q: What is the matrix forreflection in the plane3x+ 4y − 12z = 0?
Q: What is the matrix forreflection in the plane3x+ 4y − 12z = 11?
Graphics 2011/2012, 4th quarter Lecture 5: linear and affine transformations
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